An iterative method for the canard explosion in general planar systems

TL;DR

This paper presents an iterative method to identify canard explosions in general planar systems of nonlinear ODEs, extending understanding beyond singular perturbation problems, with applications demonstrated on the van der Pol equation and a self-replicating system model.

Contribution

It introduces a novel iterative approach to locate canard points in systems without small parameters, broadening the analysis of canard phenomena.

Findings

Method accurately predicts canard points in van der Pol system

Successfully applied to a templator self-replicating model

Asymptotic analysis confirms the method's validity

Abstract

The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter is present. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.